Average
1. What is an Average?
An average (also called arithmetic mean) is a value that represents the central or typical value in a set of numbers. It is calculated by dividing the sum of all values by the number of values.
Formula:
$$ \text{Average} = \frac{\text{Sum of Terms}}{\text{Number of Terms}} $$Example: If you have the numbers 10, 20, and 30:
$$ \text{Average} = \frac{10 + 20 + 30}{3} = \frac{60}{3} = 20 $$2. Why Use Averages?
Averages help us:
- Summarize large data sets with a single value.
- Compare different sets of data easily.
- Find a “fair share” or “equal distribution” among all items or people.
3. Step-by-Step: How to Calculate an Average
- Add up all the values.
- Count the number of values.
- Divide the total sum by the count.
Example: Find the average of 5, 7, 9, 11, 13.
- Sum = 5 + 7 + 9 + 11 + 13 = 45
- Number of terms = 5
- Average = 45 / 5 = 9
4. Important Average Formulas and Shortcuts
| Situation | Formula/Shortcut |
|---|---|
| Average of first $ n $ natural numbers | $ \frac{n+1}{2} $ |
| Average of first $ n $ even numbers | $ n+1 $ |
| Average of first $ n $ odd numbers | $ n $ |
| Average of consecutive numbers | $ \frac{First + Last}{2} $ |
| Average of squares of first $ n $ numbers | $ \frac{(n+1)(2n+1)}{6} $ |
| Average of cubes of first $ n $ numbers | $ \frac{n(n+1)^2}{4} $ |
| Average of $ n $ multiples of a number | $ Number \times \frac{n+1}{2} $ |
Example: Average of first 5 odd numbers: 1, 3, 5, 7, 9 Formula: $ n = 5 \Rightarrow Average = 5 $
5. Special Cases and Concepts
a) Average Speed
- For equal distances at speeds $ x $ and $ y $:
- For different distances:
Example: A car travels 60 km at 30 km/h and returns 60 km at 20 km/h.
$$ \text{Average Speed} = \frac{2 \times 30 \times 20}{30 + 20} = \frac{1200}{50} = 24\ \text{km/h} $$b) Changing Averages (Ages, Weights, etc.)
- When a new person joins or leaves a group:
- Increase in average:
- Decrease in average:
$$ \text{Age of new member} = \text{Previous average} - (\text{Decrease} \times \text{Number of members}) $$Example: Average age of 8 people increases by 2.5 years when a new person replaces one weighing 40 kg.
$$ \text{New person's weight} = 40 + (8 \times 2.5) = 40 + 20 = 60\ \text{kg} $$c) Average of Combined Groups
If two groups have averages $ a_1 $ and $ a_2 $, with $ n_1 $ and $ n_2 $ members:
$$ \text{Combined Average} = \frac{n_1 a_1 + n_2 a_2}{n_1 + n_2} $$6. Solved Examples
Example 1: Basic Average
Find the average of 65, 85, 70, 90, 105.
$$ \text{Average} = \frac{65 + 85 + 70 + 90 + 105}{5} = \frac{415}{5} = 83 $$Example 2: Average with Exclusion
The average of 5 numbers is 29. If one number is excluded, the average becomes 27. Find the excluded number.
$$ \text{Excluded number} = (5 \times 29) - (4 \times 27) = 145 - 108 = 37 $$Example 3: Average Speed (Different Distances & Speeds)
A person covers 20 km at 5 km/h, 15 km at 3 km/h, and 10 km at 2 km/h. Find the average speed.
$$ \text{Total distance} = 20 + 15 + 10 = 45\ \text{km} $$$$ \text{Time for each segment} = \frac{20}{5} + \frac{15}{3} + \frac{10}{2} = 4 + 5 + 5 = 14\ \text{hours} $$$$ \text{Average speed} = \frac{45}{14} \approx 3.21\ \text{km/h} $$Example 4: Consecutive Numbers
The average of 6 consecutive even numbers is 21. Find the largest number.
$$ \text{Largest number} = \text{Average} + (n-1) = 21 + (6-1) = 26 $$7. Practice Questions with Answers
| # | Question | Solution/Answer |
|---|---|---|
| 1 | The average of 11 observations is 90. The average of first 5 is 87, last 5 is 84. Find the 6th observation. | Sum = 11×90 = 990; First 5 = 5×87 = 435; Last 5 = 5×84 = 420; 6th = 990 - 435 - 420 = 135 |
| 2 | A batsman’s average for 12 innings is x. In 13th inning he scores 96 and his average increases by 5. Find new average. | Let old average = y; (12y + 96) / 13 = y + 5 ⇒ 12y + 96 = 13y + 65 ⇒ y = 31; New average = 36 |
| 3 | The average of 5, 10, 12, 18, 35 is? | (5 + 10 + 12 + 18 + 35) / 5 = 80 / 5 = 16 |
| 4 | Average marks of 120 candidates is 35. If average of passed is 39 and failed is 15, number passed? | Let passed = x, failed = 120 - x; 39x + 15(120-x) = 35×120 ⇒ x = 100 |
| 5 | A train travels from A to B at 20 km/h and returns at 30 km/h. What is average speed? | (2×20×30)/(20+30) = 1200/50 = 24 km/h |
8. Tips and Tricks
- For consecutive numbers, the average is always the middle number (if odd count) or mean of two middle numbers (if even count).
- For average speed, always use total distance over total time, not the arithmetic mean of speeds.
- When a group’s average changes due to a new member, use the change in total sum to find the new or replaced value.
9. Summary Table of Key Formulas
| Situation | Formula |
|---|---|
| Basic Average | $ \frac{Sum}{Count} $ |
| Average Speed (equal distance) | $ \frac{2xy}{x+y} $ |
| Average Speed (unequal distances) | $ \frac{Total Distance}{Total Time} $ |
| Average of n natural numbers | $ \frac{n+1}{2} $ |
| Average of n even numbers | $ n+1 $ |
| Average of n odd numbers | $ n $ |
10. Practice Set
Try these yourself!
- Find the average of 14, 18, 22, 26, 30.
- The average of 6 numbers is 15. If one number is 9, what is the average of the remaining 5?
- A person travels 60 km at 40 km/h and 90 km at 60 km/h. What is the average speed?
- The average age of 8 people increases by 3 years when a new person replaces one aged 20. What is the new person’s age?
- The average of 10 consecutive numbers is 23. What is the largest number?
Answers:
- (14+18+22+26+30)/5 = 110/5 = 22
- (15×6 - 9)/5 = (90 - 9)/5 = 81/5 = 16.2
- Total distance = 150 km; Total time = (60/40) + (90/60) = 1.5 + 1.5 = 3 hours; Average speed = 150/3 = 50 km/h
- New age = 20 + (8×3) = 44
- Largest = 23 + (10-1)/2 = 23 + 4.5 = 27.5 (If numbers are consecutive integers, largest is 23 + 4 = 27)
This guide covers all essential concepts, formulas, and problem types for averages, with improved explanations, real-life examples, and a variety of practice questions and answers for mastery.